A114129 -id:A114129 - cp's OEIS frontend

A164382 Take sequence A114129, those integers that are factored into prime powers each with a distinct prime exponent. If the largest power of p dividing A114129(n) is p^q(p), p and q being primes, then a(n) = product{p|A114129(n)} q(p)^p.

4, 9, 8, 32, 27, 25, 128, 72, 108, 2048, 243, 49, 8192, 288, 125, 200, 131072, 2187, 524288, 1152, 972, 8388608, 864, 800, 536870912, 675, 2147483648, 18432, 500, 1944, 392, 3456, 177147, 73728, 137438953472, 8748, 3200, 2199023255552

Offset: 1

Leroy Quet, Aug 14 2009

nonn

This is a permutation of the terms of A114129.

			288 is factored as 2^5 * 3^2. (Since the exponents 5 and 2 are distinct primes, then 288 is in sequence A114129.) The term of this sequence that corresponds to A114129(16) = 288 is then: a(16) = 5^2 * 2^3 = 200. Notably, 200 occurs in sequence A114129, as do all other terms of this sequence.

Cf. A114129.

Extended by Ray Chandler, Mar 15 2010

A087315 a(n) = Product_{k=1..n} prime(k)^prime(n-k+1).

Original entry on oeis.org

1, 4, 72, 21600, 190512000, 580909190400000, 428616352408083840000000, 859278392084450410309036800000000000, 2097197194438629126172451944256706311040000000000000

Offset: 0

Amarnath Murthy, Sep 03 2003

nonn

			a(3) = 2^5*3^3*5^2 = 21600.

G. C. Greubel, Table of n, a(n) for n = 0..23

Cf. A006939, A002110, A025487, A004394, A002093, A002182, A055932, A025487, A089247, A071364, A097320, A046523, A056166, A114129, A053810.

[1] cat [(&*[NthPrime(k)^(NthPrime(n-k+1)): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

seq(product(ithprime(k)^ithprime(n-k+1), k=1..n), n=0..10);

Table[Product[Prime[k]^Prime[n - k + 1], {k, 1, n}], {n, 0, 10}] (* G. C. Greubel, Oct 14 2018 *)

for(n=0, 10, print1(prod(k=1,n, prime(k)^prime(n-k+1)), ", ")) \\ G. C. Greubel, Oct 14 2018

[prod(nth_prime(i)^nth_prime(k-i+1) for i in (1..k)) for k in (0..10)] # Giuseppe Coppoletta, Nov 03 2014

More terms from Jorge Coveiro, Dec 22 2004

Corrected by David Wasserman, May 02 2005

A272858 Numbers m such that Product(1 + p_i) = Product(1 + e_i), where m = Product((p_i)^e_i).

Original entry on oeis.org

1, 4, 27, 72, 96, 108, 486, 800, 1280, 3125, 6272, 10976, 12500, 14336, 21600, 28800, 30375, 34560, 36000, 38880, 48600, 54675, 84375, 92160, 96000, 121500, 134456, 153600, 169344, 217728, 218700, 225000, 247808, 262440, 296352, 300000, 337500, 340736, 387072, 395136, 489888, 666792, 703125, 750141, 781250, 823543, 857304, 885735

Offset: 1

Giuseppe Coppoletta, May 08 2016

nonn

A048102 is clearly a subsequence, as for any prime p, p^p satisfy the herein condition. Similarly, A122406 is also a subsequence. More generally, if a number is a term, then any permutation of the exponents in its prime factorization (i.e., any permutation of its prime signature) gives also a term.

The condition defining this sequence coincides with the condition in A272859 at least for the terms of A114129.

			92160 is included because 92160 = 2^11 * 3^2 * 5 and (2+1)*(3+1)*(5+1) = (11+1)*(2+1)*(1+1).

Giuseppe Coppoletta and Giovanni Resta, Table of n, a(n) for n = 1..6058 (terms < 10^18, first 100 terms from G. Coppoletta)

Cf. A048102, A054411, A054412, A071174, A114129, A122406, A272818, A272859.

ok[n_] := Block[{p,e}, {p,e} = Transpose@ FactorInteger@ n; Times @@ (1+p) == Times @@ (1+e)]; Select[Range[10^6], ok] (* Giovanni Resta, May 08 2016 *)

is(n)=my(f=factor(n)); prod(i=1,#f~, f[i,1]+1)==prod(i=1,#f~,f[i,2]) \\ Charles R Greathouse IV, Sep 08 2016

def d(n):
    v = factor(n)
    d1 = prod(1 + w[0] for w in v)
    d2 = prod(1 + w[1] for w in v)
    return d1 == d2
[k for k in (1..10000) if d(k)]

If N is a positive integer and N = Product_{i=1..k} (p_i)^e_i is its prime factorization, then N is in A272858 iff Product_{i=1..k} (1 + p_i) = Product_{i=1..k} (1 + e_i).

For a number with three different prime factors N = p1^e1 * p2^e2 * p3^e3, the defining condition can be expressed as: p1 + p2 + p3 + p1*p2 + p1*p3 + p2*p3 + p1*p2*p3 = e1 + e2 + e3 + e1*e2 + e1*e3 + e2*e3 + e1*e2*e3.

A272859 Numbers m such that sigma(Product(p_j)) = sigma(Product(e_j)), where m = Product((p_i)^e_i) and sigma = A000203.

Original entry on oeis.org

1, 4, 27, 72, 108, 192, 800, 1458, 3125, 5120, 6144, 6272, 10976, 12500, 21600, 30375, 36000, 48600, 54675, 77760, 84375, 114688, 116640, 121500, 134456, 138240, 169344, 173056, 225000, 229376, 247808, 337500, 354294, 384000, 395136, 600000, 653184, 655360, 703125, 750141, 823543, 857304, 913952, 979776

Offset: 1

Giuseppe Coppoletta, May 08 2016

nonn

A048102 is clearly a subsequence, as for any prime p, p^p satisfy the herein condition. Moreover, due to the multiplicativity of the arithmetic function sigma, A122406 is also a subsequence. More generally, if a number is a term, then any permutation of the exponents in its prime factorization (i.e., any permutation of its prime signature) gives also a term.

The condition defining this sequence coincides with the condition in A272858 at least for the terms of A114129.

			173056 is included because 173056 = 2^10 * 13^2 and sigma(2*13) = sigma(10*2).
653184 is included because 653184 = 2^7 * 3^6 * 7 and sigma(2*3*7) = sigma(7*6*1).

Giuseppe Coppoletta and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 100 terms from G. Coppoletta)

Cf. A048102, A054411, A054412, A071174, A114129, A122406, A272818, A272858.

Select[Range[10^6], First@ # == Last@ # &@ Map[DivisorSigma[1, Times @@ #] &, Transpose@ FactorInteger@ #] &] (* Michael De Vlieger, May 12 2016 *)

A272859 = []
for n in (1..10000):
    v = factor(n)
    if prod(1 + w[0] for w in v) == sigma(prod(w[1] for w in v)): A272859.append(n)
print(A272859)

A114128 Numbers that factorize into a prime number of distinct prime factors each raised to a different prime exponent.

Original entry on oeis.org

72, 108, 200, 288, 392, 500, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1944, 2312, 2888, 3087, 3200, 3267, 3456, 3872, 4000, 4232, 4563, 5324, 5408, 6075, 6125, 6272, 6728, 7688, 7803, 8575, 8748, 8788, 9248, 9747, 10952, 10976

Offset: 1

Jon Wild, Feb 14 2006

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..600

Subsequence of A114129.

pnpfQ[n_]:=Module[{pn=PrimeNu[n],fi=Transpose[FactorInteger[n]][[2]]}, PrimeQ[ pn]&&Length[Union[fi]]==pn&&AllTrue[fi,PrimeQ]]; Select[Range[ 11000], pnpfQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 12 2015 *)

isok(n) = {nbf = omega(n); if (! isprime(nbf), return (0)); f = factor(n); for (i = 1, nbf, if (! isprime(f[i, 2]), return (0)); for (j = i+1, nbf, if (f[i, 2] == f[j, 2], return (0)););); return (1);} \\ Michel Marcus, Aug 18 2013

Definition clarified by Harvey P. Dale, Aug 12 2015

A385288 Numbers with a prime number of prime factors, counted with multiplicity, and whose prime factors are each raised to a prime exponent.

Original entry on oeis.org

4, 8, 9, 25, 27, 32, 49, 72, 108, 121, 125, 128, 169, 200, 243, 288, 289, 343, 361, 392, 500, 529, 675, 800, 841, 961, 968, 972, 1125, 1323, 1331, 1352, 1369, 1372, 1568, 1681, 1800, 1849, 2048, 2187, 2197, 2209, 2312, 2700, 2809, 2888, 3087, 3125, 3267, 3481

Offset: 1

James C. McMahon, Jun 24 2025

nonn

a(n) = A114129(n) through n=25; then a(26) = 961 and A114129(26) = 864.
Subset of A056166.
Subset of A001694. - Michael De Vlieger, Jun 25 2025.

			200 = 2^3 * 5^2; 200 has a prime number of prime factors, counted with multiplicity (3 + 2 = 5), and exponents 3 and 2 are prime.

James C. McMahon, Table of n, a(n) for n = 1..1000

Cf. A001248, A001694, A030078, A056166, A114129.

Select[Range[10^4],AllTrue[Last/@FactorInteger[#],PrimeQ]&&PrimeQ[PrimeOmega[#]]&]

isok(k) = my(f=factor(k)); isprime(bigomega(k)) && (sum(k=1, #f~, isprime(f[k,2])) == omega(f)); \\ Michel Marcus, Jun 25 2025

cp's OEIS Frontend

A164382 Take sequence A114129, those integers that are factored into prime powers each with a distinct prime exponent. If the largest power of p dividing A114129(n) is p^q(p), p and q being primes, then a(n) = product{p|A114129(n)} q(p)^p.

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A087315 a(n) = Product_{k=1..n} prime(k)^prime(n-k+1).

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A272858 Numbers m such that Product(1 + p_i) = Product(1 + e_i), where m = Product((p_i)^e_i).

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A272859 Numbers m such that sigma(Product(p_j)) = sigma(Product(e_j)), where m = Product((p_i)^e_i) and sigma = A000203.

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Comments

Examples

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Mathematica

Sage

A114128 Numbers that factorize into a prime number of distinct prime factors each raised to a different prime exponent.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A385288 Numbers with a prime number of prime factors, counted with multiplicity, and whose prime factors are each raised to a prime exponent.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI